The correct answer is option 1, Binodini Dasi. Binodini Dasi was a pioneering Bengali woman who serialised her autobiography, Amar Katha, between 1910 and 1913. Amar Katha is a significant piece of literature as it provides insights into Binodini Dasi`s life, her experiences, and the challenges she faced as a woman in the early 20th century.

Option 2, Rashsundari Debi, is not the correct answer. Rashsundari Debi was another pioneering Bengali woman who wrote an autobiography titled Amar Jiban. However, her autobiography was not serialized between 1910 and 1913.

Option 3, Begum Rokeya Sakhawat Hossain, is also not the correct answer. Begum Rokeya Sakhawat Hossain was a notable Bengali writer, but she did not serialize her autobiography during the specified time period.

Option 4, Kailashbashini Debi, is not the correct answer. There is limited information available about Kailashbashini Debi, and she did not serialize her autobiography during the years mentioned in the question.

Therefore, the answer is option 1, Bin